Put numbers 1..64 on the chessboard in such a way that neighbors (common side) differ at most by 4

put 10 integers around a circle in such a way that each of them is an arithmetic mean of its two neighbors, and not all numbers are the same

is it possible to put numbers 1 2 3 4 5 6 7 8 around the circle in such a way that (a) sum of any two neighbors is odd; (b) sum of neighbors is even; (c) the same question for 1 2 3 4 5 6 7 and odd sum; (d) for 1 2 3 4 5 6 7 and even sum

a turtle is going forward one unit, then turns right or left (by 90 degrees), moves one unit, turns again etc. Can it return to original position after 15/16/17/18 moves?

is it possible to place signs in the expression 129 to get result 0? 1? 2? 100?

there are 5 objects in a row labeled a,b,c,d,e,f. The goal is to put them in a different order using given number of transpositions (by mouse?), or declare that this is not possible. Two configurations that should be achieved by 20/21 transpositions (if this is not possible, this should be declared).

the same as previous but only transposition of neighbors are allowed. (One of the required permutations is a transposition of two non-neighbor elements.)

Course 2: Combinatorics?

Course 3: Graphs

Introduction

In fact, it is possible

Bingo!

Congratulations! You're right!

In fact, a better solution exists

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